'Last One Standing' printed from http://nrich.maths.org/

Why do this problem?

This problem is one of a set of problems about probability and
uncertainty. Intuition can often let us down when working on
probability; these problems have been designed to provoke
discussions that challenge commonly-held misconceptions. Read more
in this
article.

Unusual events are expected to happen to someone if the population
is large enough. It's impossible to predict in advance who it will
happen to, but after the event we shouldn't be surprised that it
has occured. This problem provides an example of this
phenomenon.

Possible approach

Before the lesson, prepare a sealed envelope with the word
FIVE written on it (assuming you have a class of about 30).

Ask the class to stand up and to each flip a coin. Ask people
who flipped tails to sit down. Ask people who flipped heads to flip
again. Repeat until there's only one person standing.

Ask the last person standing how many heads they flipped, and
reveal with a flourish your prediction - there's a good chance that
it will be right, or very close!

"How did I know the last person
standing would flip about five heads in a row?"

"Why didn't I try to predict who it
would be?"

Give the class a short while to think about these questions
and then discuss them with their partner, and then the whole
class.

"Imagine we repeated this exercise in the school hall with
around 250 students (or 1000). How many heads do you think the last
one standing would have flipped?"

Give the class a short while to think about this question
before asking them to justify their predictions.

Then use the animation (set up for 256 or 1024) to test out
their suggestions - this is a good opportunity to alert learners to
the fact that a single trial will not always reflect the
theoretical probability, and to discuss the importance of
repeating an experiment and taking an average when working with
experimental probability.

There are lots of discussions that can come from this task and from
watching the animation at each stage. For example:

Why does the number of people standing halve at each
stage?

Is it possible to predict where the last person standing will
be on the grid?

The animation could be used with each learner choosing a point or
region of the grid and seeing if they chose correctly, to capture
the idea that the unexpected will happen to someone almost every
time, but that it's difficult to predict who it will happen
to.

There are some suggested questions at the end of the problem that
could be used to explore the ideas further. Alternatively, the
class could be asked to think of other examples where very unlikely
events happen in very large populations.

Key questions

What proportion of the people standing do we expect to sit
down on each flip of the coin?

Can we predict how often we should expect an event to
occur?

Can we predict to whom we expect the event to happen?

Possible extension

Same
Number! offers another opportunity to work with probabilities
in order to explain unexpected events.

Students might be interested in this article
and related materials on the Understanding Uncertainty
website.

Possible
support

Encourage learners to start by analysing what happens with only a
very small number of people in the room, and to use the
interactivity to model it.